nLab supergeometry

Idea

Supergeometry is the (higher) geometry over the base topos on superpoints modeled on the canonical line object $\mathbb{R}$ in there.

As ordinary differential geometry studies spaces – smooth manifolds – that locally look like vector spaces, supergeometry studies spaces – supermanifolds – that locally look like super vector spaces.

As ordinary algebraic geometry studies spaces – schemes – that locally look like affine spaces, supergeometry studies superschemes.

From the point of view of noncommutative geometry, the supergeometry is a very mild special case of noncommutativity in geometry: some coordinates commute, some anticommute.

For more see at geometry of physics – supergeometry.

Supergeometry may defined over an arbitrary commutative ring (Schwarz and Shapiro 06).

Related concepts

• signs in supergeometry

• superpoint

• superspace

  • supermanifold, super scheme

  • spectral super-scheme

• supersymmetry

• super-spacetime, super-Minkowski spacetime, super Poincare group

• super-vielbein

• supergravity

• Super Gerbes

• bosonic modality

• super-Cartan geometry

  • superconformal geometry

geometries of physics

$\phantom{A}$(higher) geometry$\phantom{A}$	$\phantom{A}$site$\phantom{A}$	$\phantom{A}$sheaf topos$\phantom{A}$	$\phantom{A}$∞-sheaf ∞-topos$\phantom{A}$
$\phantom{A}$discrete geometry$\phantom{A}$	$\phantom{A}$Point$\phantom{A}$	$\phantom{A}$Set$\phantom{A}$	$\phantom{A}$Discrete∞Grpd$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$CartSp$\phantom{A}$	$\phantom{A}$SmoothSet$\phantom{A}$	$\phantom{A}$Smooth∞Grpd$\phantom{A}$
$\phantom{A}$formal geometry$\phantom{A}$	$\phantom{A}$FormalCartSp$\phantom{A}$	$\phantom{A}$FormalSmoothSet$\phantom{A}$	$\phantom{A}$FormalSmooth∞Grpd$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$SuperFormalCartSp$\phantom{A}$	$\phantom{A}$SuperFormalSmoothSet$\phantom{A}$	$\phantom{A}$SuperFormalSmooth∞Grpd$\phantom{A}$

$\,$

duality between $\;$algebra and geometry

$\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$category$\phantom{A}$	$\phantom{A}$dual category$\phantom{A}$	$\phantom{A}$algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/Gelfand-Kolmogorov+theorem">Gelfand-Kolmogorov</a>}}{\hookrightarrow} Alg^{op}_{\mathbb{R}}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$topology$\phantom{A}$	$\phantom{A}$$\phantom{NC}TopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a class="existingWikiWord" href="https://ncatlab.org/nlab/show/Gelfand+duality">Gelfand duality</a>}}{\simeq} TopAlg^{op}_{C^\ast, comm}$$\phantom{A}$	$\phantom{A}$comm. C-star-algebra$\phantom{A}$
$\phantom{A}$noncomm. topology$\phantom{A}$	$\phantom{A}$$NCTopSpaces_{H,cpt}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} TopAlg^{op}_{C^\ast}$$\phantom{A}$	$\phantom{A}$general C-star-algebra$\phantom{A}$
$\phantom{A}$algebraic geometry$\phantom{A}$	$\phantom{A}$$\phantom{NC}Schemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/affine+scheme#AffineSchemesFullSubcategoryOfOppositeOfRings">almost by def.</a>}}{\simeq} \phantom{Top}Alg^{op}$$\phantom{A}$	$\phantom{A} \phantom{A}$ $\phantom{A}$commutative ring$\phantom{A}$
$\phantom{A}$noncomm. algebraic$\phantom{A}$ $\phantom{A}$geometry$\phantom{A}$	$\phantom{A}$$NCSchemes_{Aff}$$\phantom{A}$	$\phantom{A}$$\overset{\phantom{\text{Gelfand duality}}}{\coloneqq} \phantom{Top}Alg^{op}_{fin, red}$$\phantom{A}$	$\phantom{A}$fin. gen. $\phantom{A}$associative algebra$\phantom{A}$$\phantom{A}$
$\phantom{A}$differential geometry$\phantom{A}$	$\phantom{A}$$SmoothManifolds$$\phantom{A}$	$\phantom{A}$$\overset{\text{<a href="https://ncatlab.org/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">Milnor's exercise</a>}}{\hookrightarrow} \phantom{Top}Alg^{op}_{comm}$$\phantom{A}$	$\phantom{A}$commutative algebra$\phantom{A}$
$\phantom{A}$supergeometry$\phantom{A}$	$\phantom{A}$$\array{SuperSpaces_{Cart} \\ \\ \mathbb{R}^{n\vert q}}$$\phantom{A}$	$\phantom{A}$$\array{ \overset{\phantom{\text{Milnor's exercise}}}{\hookrightarrow} & Alg^{op}_{\mathbb{Z}_2 \phantom{AAAA}} \\ \mapsto & C^\infty(\mathbb{R}^n) \otimes \wedge^\bullet \mathbb{R}^q }$$\phantom{A}$	$\phantom{A}$supercommutative$\phantom{A}$ $\phantom{A}$superalgebra$\phantom{A}$
$\phantom{A}$formal higher$\phantom{A}$ $\phantom{A}$supergeometry$\phantom{A}$ $\phantom{A}$(super Lie theory)$\phantom{A}$	$\phantom{A}\array{ Super L_\infty Alg_{fin} \\ \mathfrak{g} }\phantom{A}$	$\phantom{A}\array{ \overset{ \phantom{A}\text{<a href="https://ncatlab.org/nlab/show/L-infinity-algebra#ReformulationInTermsOfSemifreeDGAlgebra">Lada-Markl</a>}\phantom{A} }{\hookrightarrow} & sdgcAlg^{op} \\ \mapsto & CE(\mathfrak{g}) }\phantom{A}$	$\phantom{A}$differential graded-commutative$\phantom{A}$ $\phantom{A}$superalgebra $\phantom{A}$ (“FDAs”)

in physics:

$\phantom{A}$algebra$\phantom{A}$	$\phantom{A}$geometry$\phantom{A}$
$\phantom{A}$Poisson algebra$\phantom{A}$	$\phantom{A}$Poisson manifold$\phantom{A}$
$\phantom{A}$deformation quantization$\phantom{A}$	$\phantom{A}$geometric quantization$\phantom{A}$
$\phantom{A}$algebra of observables	$\phantom{A}$space of states$\phantom{A}$
$\phantom{A}$Heisenberg picture	$\phantom{A}$Schrödinger picture$\phantom{A}$
$\phantom{A}$AQFT$\phantom{A}$	$\phantom{A}$FQFT$\phantom{A}$
$\phantom{A}$higher algebra$\phantom{A}$	$\phantom{A}$higher geometry$\phantom{A}$
$\phantom{A}$Poisson n-algebra$\phantom{A}$	$\phantom{A}$n-plectic manifold$\phantom{A}$
$\phantom{A}$En-algebras$\phantom{A}$	$\phantom{A}$higher symplectic geometry$\phantom{A}$
$\phantom{A}$BD-BV quantization$\phantom{A}$	$\phantom{A}$higher geometric quantization$\phantom{A}$
$\phantom{A}$factorization algebra of observables$\phantom{A}$	$\phantom{A}$extended quantum field theory$\phantom{A}$
$\phantom{A}$factorization homology$\phantom{A}$	$\phantom{A}$cobordism representation$\phantom{A}$

References

General

Historically influential general considerations:

• Yuri Manin, New Dimensions in Geometry, in: Arbeitstagung Bonn 1984, Lecture Notes in Mathematics 1111, Springer (1985) 59–101 [doi:10.1007/BFb0084585, pdf]

• Felix A. Berezin (edited by Alexandre A. Kirillov): Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6]

Survey:

• Yuri Manin, Chapter 4 in: Gauge Field Theory and Complex Geometry, Grundlehren der Mathematischen Wissenschaften 289, Springer (1988) [doi:10.1007/978-3-662-07386-5]

• John Lott, Torsion constraints in supergeometry, Comm. Math. Phys. 133 (1990) 563-615 [doi:10.1007/BF02097010]

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.2 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, chII.2: pdf]

• Pierre Deligne, John Morgan, Notes on Supersymmetry (following Joseph Bernstein), in: Quantum Fields and Strings, A course for mathematicians, 1, Amer. Math. Soc. Providence (1999) 41-97 [ISBN:978-0-8218-2014-8, web version, pdf]

• Ivan Mirković, Notes on Super Math, in Quantum Field Theory Seminar, lecture notes (2004) [pdf, pdf]

Introductory lecture notes:

• Daniel Freed, Five lectures on supersymmetry

• Gennadi Sardanashvily, Lectures on supergeometry [arXiv:0910.0092, inspire:832789]

Discussion via functorial geometry:

• Schmitt 1997

• Claudio Carmeli, Lauren Caston, Rita Fioresi: Mathematical foundations of supersymmetry, EMS Series of Lectures in Mathematics, EMS (2011) [ems:elm/97, doi:10.4171/097, epdf, short draft: arXiv:0710.5742]

Lectures on supergeometry highlighting methods from algebraic geometry:

• Pierre Deligne: Supermoduli – Methods from Algebraic Geometry, lecture series at Supermoduli Workshop, Simons Center @ Stony Brook (2015) [scgp video: I, II, III, IV, V; YouTube:I]

Discussion of traditional algebraic geometry for super-schemes:

• Ugo Bruzzo, Daniel Hernandez Ruiperez, Alexander Polishchuk, Notes on fundamental algebraic supergeometry. Hilbert and Picard superschemes (arXiv:2008.00700)

The observation that the study of super-structures in mathematics is usefully regarded as taking place over the base topos on the site of super points has been made around 1984 in

• Albert Schwarz, On the definition of superspace, Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 37–42, (russian original pdf)

• Alexander Voronov, Maps of supermanifolds , Teoret. Mat. Fiz. (1984) Volume 60, Number 1, Pages 43–48

and in

• V. Molotkov, Infinite-dimensional $\mathbb{Z}_2^k$-supermanifolds, ICTP preprints, IC/84/183, 1984.

A summary/review is in the appendix of

• Anatoly Konechny, Albert Schwarz,

  On $(k \oplus l|q)$-dimensional supermanifolds, in Julius Wess, V. Akulov (eds.) Supersymmetry and Quantum Field Theory (Dmitry Volkov memorial volume), Lecture Notes in Physics 509, Springer (1998) [arXiv:hep-th/9706003, doi:10.1007/BFb0105247]

  Theory of $(k \oplus l|q)$-dimensional supermanifolds Sel. math., New ser. 6 (2000) 471 - 486

Generalization to supergeometry over an arbitrary commutative ring, in particular $p$-adic rings, is given in

• Albert Schwarz, I. Shapiro, Supergeometry and Arithmetic Geometry (arXiv:hep-th/0605119)

A review of all this as geometry in the topos over the category of superpoints is in

• Christoph Sachse, A Categorical Formulation of Superalgebra and Supergeometry [arXiv:0802.4067]

Formulation in terms of synthetic differential supergeometry is in

• David Carchedi, Dmitry Roytenberg, On theories of superalgebras of differentiable functions, Theory and Applications of Categories, Vol. 28, 2013, No. 30, pp 1022-1098. (arxiv:1211.6134, TAC)

For many more references see at supermanifold.

Plenty of discussion of supergeometry with an eye towards supersymmetry in quantum field theory is in

• P. Deligne, P. Etingof, D.S. Freed, L. Jeffrey, D. Kazhdan, J. Morgan, D.R. Morrison, E. Witten (eds.) Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence 1999. (web version)

especially in the contribution

• Pierre Deligne, Daniel Freed, Supersolutions (arXiv:hep-th/9901094).

The appendix there

• Sign manifesto (pdf)

means to sort out various sign conventions of relevance.

Discussion of how supersymmetry is universally induced in higher category theory/homotopy theory by the free abelian ∞-group on the point – the sphere spectrum – is in

• Mikhail Kapranov, Categorification of supersymmetry and stable homotopy groups of spheres, April 2013 (abstract, video)

For more on this see at superalgebra.

Discussion related to G-structure and Killing spinors includes

• Dmitri Alekseevsky, Vicente Cortés, Chandrashekar Devchand, Uwe Semmelmann, Killing spinors are Killing vector fields in Riemannian Supergeometry (arXiv:dg-ga/9704002)

Generalization of Artin's representability theorem to supergeometry:

• Nadia Ott, Artin’s theorems in supergeometry (arXiv:2110.12816)

For AQFT-like discussion of supersymmetric field theory:

• Thomas-Paul Hack, Florian Hanisch, Alexander Schenkel: Section 3 of: Supergeometry in locally covariant quantum field theory, Commun. Math. Phys. 342 615 (2016) [doi:10.1007/s00220-015-2516-4, arXiv:1501.01520]

See also:

• Johannes Moerland: Spinorial Superspaces and Super Yang-Mills Theories [arXiv:2411.06165]

Supergeometry of fermion fields

Discussion of the classical mechanics of the spinning particle or of classical field theory with fermion fields (possibly but not necessarily super-symmetric) as taking place in supergeometry:

via (possibly infinite-dimensional) supermanifolds:

• Felix A. Berezin, M. S. Marinov: Particle Spin Dynamics as the Grassmann Variant of Classical Mechanics, Annals of Physics 104 2 (1977) 336-362 [doi:10.1016/0003-4916(77)90335-9, pdf, pdf]

  reprinted in Appendix I of: Alexandre A. Kirillov (ed.): Introduction to Superanalysis, Mathematical Physics and Applied Mathematics 9, Springer (1987) [doi:10.1007/978-94-017-1963-6]

• Thomas Schmitt: The Cauchy Problem for Classical Field Equations with Ghost and Fermion Fields [arXiv:hep-th/9607133]

• Thomas Schmitt: Supergeometry and Quantum Field Theory, or: What is a Classical Configuration?, Rev. Math. Phys. 9 (1997) 993-1052 [doi:10.1142/S0129055X97000348, arXiv:hep-th/9607132].

• Thomas Schmitt: Supermanifolds of classical solutions for Lagrangian field models with ghost and fermion fields, Sfb 288 Preprint No. 270 [hep-th/9707104, inspire:445574]

• Daniel Freed, What are fermions?, Lecture 1 in: Five lectures on supersymmetry, AMS (1999) [ISBN:978-0-8218-1953-1, spire:517862]

• Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily, chapter 3 of: Advanced classical field theory, World Scientific (2009) [doi:10.1142/7189]

• Gennadi Sardanashvily, Grassmann-graded Lagrangian theory of even and odd variables, [arXiv:1206.2508]

• Gennadi Sardanashvily W. Wachowski: SUSY gauge theory on graded manifolds [arXiv:1406.6318, spire:1302860]

• Viola Gattus, Apostolos Pilaftsis, Supergeometric Approach to Quantum Field Theory, CORFU2023, PoS 463 (2024) 156 [doi:10.22323/1.463.0156, arXiv:2404.13107]

• Viola Gattus, Apostolos Pilaftsis: Supergeometric Quantum Effective Action [arXiv:2406.13594]

and more generally via smooth super sets:

• Urs Schreiber: §3.1 in: Higher Prequantum Geometry, chapter in: New Spaces for Mathematics and Physics, Cambridge University Press (2021) [doi:10.1017/9781108854399.008, arXiv:1601.05956]

Discussion with focus on supersymmetry:

• Leonardo Castellani, Riccardo D'Auria, Pietro Fré, section II.2.4 of: Supergravity and Superstrings - A Geometric Perspective, World Scientific (1991) [doi:10.1142/0224, toc: pdf, chII.2: pdf]

• Pierre Deligne, Daniel Freed: Supersolutions, in: Quantum Fields and Strings, A course for mathematicians, 2 vols. Amer. Math. Soc. Providence (1999) 357-366 [arXiv:hep-th/9901094, ISBN:978-0-8218-2014-8, web version]

• Daniel Freed, Classical field theory and Supersymmetry, IAS/Park City Mathematics Series 11 (2001) [pdf, pdf]

and specifically in the context of super- string theory (regarding worldsheets as super Riemann surfaces):

• Eric D'Hoker, Duong Phong, Complex geometry and supergeometry, Current Developments in Mathematics 2005 (2007) 1-40 [arXiv:hep-th/0512197, euclid.cdm/1223654523]